2011 AMC 10B Problem 4

P1-10~1 minPrint

Linear Equations & Word Problems Set up the equation/formula and compute; no special trick needed

LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip, it turned out that LeRoy had paid $A$ dollars and Bernardo had paid $B$ dollars, where $A < B$ . How many dollars must LeRoy give to Bernardo so that they share the costs equally?

Show solution without answering

Key insight

Each person should pay half of the total. LeRoy gives Bernardo the difference between that half and what he already paid.

Solution

Try real numbers first to see what happens. Say LeRoy paid $A = 10$ dollars and Bernardo paid $B = 30$ dollars.

The trip cost $10 + 30 = 40$ dollars, so each should pay $20$ .
LeRoy paid only $10$ , so he owes $20 - 10 = 10$ more. He gives Bernardo $10$ .
Check: Bernardo paid $30$ and gets $10$ back, so he paid $20$ too.

Now do the same steps with letters.

4. The total is $A + B$ , so each person's fair share is $\dfrac{A + B}{2}$ .
5. LeRoy has paid $A$ , so he still owes
$\frac{A + B}{2} - A = \frac{A + B - 2A}{2} = \frac{B - A}{2}.$
6. With the test numbers, this gives $\frac{30 - 10}{2} = 10$ , which matches.

The answer is $\boxed{\textbf{(C)}\ \dfrac{B - A}{2}}$ .

Why this works

To make two amounts equal, you move half of the gap between them from the bigger one to the smaller one. Moving the whole gap would just swap which person is ahead. Testing easy numbers like $10$ and $30$ is a quick way to check any answer written with letters.

Alternative approach

Plug $A = 10$ and $B = 30$ into every choice and look for $10$ . (A) gives $20$ , (B) gives $-10$ , (C) gives $10$ , (D) gives $20$ , (E) gives $40$ . Only (C) matches.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

The trip cost $A + B$ dollars in total, so each person's fair share is $\dfrac{A + B}{2}$ .

LeRoy has paid only $A$ , so he owes
$\frac{A + B}{2} - A = \frac{B - A}{2}.$
This is positive because $A < B$ , and after the transfer Bernardo's net payment is $B - \frac{B - A}{2} = \frac{A + B}{2}$ as well.

The answer is $\boxed{\textbf{(C)}\ \dfrac{B - A}{2}}$ .

Review in the AoPS books

Introduction to Algebra, Chapter 3: One-Variable Linear Equations (§3.3, §3.4)
Introduction to Algebra, Chapter 7: Proportion (§7.4)